# A case study on the use of scale separation-based analytic propagators for parameter inference in stochastic gene regulation

## Summary (3 min read)

### 1. INTRODUCTION

- Gene expression is a complex and highly regulated multi-step process that is responsible for the timely synthesis of proteins necessary for cellular function.
- A simple, “two-stage” model for stochastic gene expression consists of a constitutively active gene from which an mRNA molecule can be transcribed, and protein, the production of which depends on the instantaneous abundance of mRNA (see Fig. 1A).
- In recent work [14], the procedure developed in [2] was extended to capture departure from the assumption of perfect scale separation: the ratio of degradation rates of protein and mRNA, denoted ε, was taken to be small and positive instead of zero, as was the case in [2].
- While fluorescence microscopy yields only time-series for the intensity, these can nonetheless be converted into absolute protein numbers if a calibration factor of molecules per unit intensity can be estimated, see e.g. [19].
- For comparison, both propagators are also contrasted with an approximate solution of the CME that is computed using a finite state projection.

### 2.1. Two-stage Gene Expression Model

- The authors model gene expression as a two-stage process, whereby DNA is transcribed to mRNA, which is then translated into protein (see Fig. 1A).
- (1) Here,m and n denote mRNA and protein copy numbers, respectively, a is the non-dimensional transcription rate and b is the non-dimensional translation rate, while the degradation rates of mRNA and protein are given by γ and 1, respectively (cf. Fig. 1A).
- Finally, τ denotes a suitably nondimensionalized time variable.
- It follows that for ε sufficiently small, the dynamics of Eq. (1) will vary on two distinct time-scales: the long-term behavior of the system is naturally described on the “slow” τ -scale, while the “fast” transients evolve according to the rescaled time t := τε .

### 2.2.2. Uniform (First-Order) Propagator

- Here, ε denotes the per- turbation parameter, as before, while t is the fast time variable.
- The authors remark that the transition probability Pn|n0(τ, ε) contributes on the slow τ -scale in Eq. (3), while the t-dependent contribution in Eq. (3) accounts for the transient dynamics on the fast time-scale.

### 2.3. Special Cases of the Hypergeometric Functions

- Care must be taken when evaluating the hypergeometric function 2F1(a, b, c, z).
- The following special cases are of use [20].

### 2.4. Stochastic Simulation

- Stochastic simulations were performed using the StochKit 2.0 [21] simulation framework and the standard stochastic simulation algorithm [3], with a non-dimensionalized transcription rate a = 20 and a non-dimensionalized translation rate b = 2.5, corresponding to “regime I” as defined in [14].
- Each value of γ was simulated 20 times, and the resulting trajectories were used for computing the probability landscapes of the rescaled model parameters a and b.
- Protein quantities were observed without measurement noise at intervals of 0.1 time units.
- All simulation runs assumed zero molecules of mRNA and protein initially, i.e., m0 = 0 = n0.

### 2.5. Implementation

- Both the zeroth-order propagator Pn|n0 , Eq. (2), and the uniform propagator Pn|n0 , Eq. (3), were implemented in C++ with a Matlab mex-file interface.
- Special functions were evaluated using the GNU scientific library [22], the Hyp_2F1 function implementation of the Gauss hypergeometric function [23], and the Algorithm 910multiprecision special function library [24].
- While the difference of such numbers is potentially below a double precision machine error of approximately 10−13, they are nonetheless essential in the correct computation of the transition probabilities.
- Their C++ implementation is still inaccurate in some extreme cases, typically for very large protein numbers n, due to numerical differences which are sometimes as small as 10−370 in Eq. (7), but which unfortunately cannot be neglected as they are inflated by the remaining terms in the expression.
- Such inaccuracies are infrequent, though, and generally occur during transitions for which the uniform propagator yields nonphysical values; thus, they do not substantially affect their analysis, or the conclusions obtained in this study.

### 3. RESULTS AND DISCUSSION

- To assess the applicability of the zeroth-order propagator Pn|n0(τ, 0), Eq. (2), and the uniform propagator Pn|n0(τ, t, ε), Eq. (3), for parameter inference in the two-stage gene expression model, the authors simulated time-series with a specific parameter pair (a∗, b∗).
- Then, the authors computed the likelihood of the observed data set on the basis of the two propagators for a range of values for the parameters a and b.
- For simplicity, the authors assumed the scale separation γ between mRNA and protein lifetimes to be known (see Methods for definitions).

### 3.2. Parameter Inference

- The term ni represents the number of proteins at measurement time ti.
- Thus, the authors compute the logarithm of the probability of each transition, from ni−1 protein molecules at time ti−1 to ni molecules at time ti, in the sequence of observed measurements (see Fig. 1B, inset).
- In order to estimate the parameters a and b from simulated protein time-courses, Eq. (13) has to be evaluated very frequently.
- The authors thus developed a numerically stable expression for the uniform propagator Pn|n0 (see Section 2 2.2), and they used an efficient implementation in C++ for both propagators that results in reasonable runtimes; see Section 2 2.5 for details.

### 3.3. Comparison of Propagator Accuracy and Efficiency

- For each pair (a, b), the authors computed the log-likelihood L(a, b), thus obtaining a likelihood landscape that should ideally have its maximum, the maximum likelihood estimator (MLE), at the true parameter values (a∗, b∗).
- The authors immediately encountered the obstacle that the uniform propagator Pn|n0 yields negative transition probabilities, or even probabilities larger than one, for some choices of (a, b).
- The resulting non-physicality is discussed in [14], and is due to the fact that Pn|n0 is derived from an asymptotic approximation; nonetheless, it is problematic when computing the overall log-likelihood, as the definition in Eq. (13) becomes meaningless.
- In Fig. 3B, a typical protein time-course with γ = 10 is shown (top), along with the log-likelihood obtained from the uniform propagator Pn|n0 , Eq. (3), for the true parameter values , and for the MLE (cyan).
- From that plot, it is obvious that large regions of the transition space yield non-physical values, shown in gray.

### 4. CONCLUSION

- The authors have investigated the utility of a propagator-based approach for approximating the transition probabilities in a simple two-stage gene expression model by attempting parameter inference from protein time-series.
- The simulations were initialized with zero molecules of both mRNA and protein; this simplification, as compared to a typical biological setting, does not affect the subsequent analysis.
- Alternatively, one could use Markov Chain Monte Carlo (MCMC) techniques to sample directly from the posterior in order to obtain the log-likelihood landscape [26].
- The variance of the noise then constitutes an additional unknown parameter σ which would have to be inferred.

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